Factor the following expression: $8$ $x^2$ $-31$ $x$ $-45$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(8)}{(-45)} &=& -360 \\ {a} + {b} &=& & & {-31} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-360$ and add them together. Remember, since $-360$ is negative, one of the factors must be negative. The factors that add up to ${-31}$ will be your ${a}$ and ${b}$ When ${a}$ is ${9}$ and ${b}$ is ${-40}$ $ \begin{eqnarray} {ab} &=& ({9})({-40}) &=& -360 \\ {a} + {b} &=& {9} + {-40} &=& -31 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {8}x^2 +{9}x {-40}x {-45} $ Group the terms so that there is a common factor in each group: $ ({8}x^2 +{9}x) + ({-40}x {-45}) $ Factor out the common factors: $ x(8x + 9) - 5(8x + 9) $ Notice how $(8x + 9)$ has become a common factor. Factor this out to find the answer. $(8x + 9)(x - 5)$